Introduction

The system given by the following schematic

has the closed loop expression:

Now suppose we wish the closed loop to respond in a certain manner, e.g. G_m(z). In other words, it is desired that the closed loop has the dynamics of G_m(z).

Then the controller that will give this closed loop response characteristic is that which satisfies the following equation:

This so called Synthesis Equation is obtained by re-arranging the closed-loop transfer function and replacing by G_m(z).

Thus, the required controller can be designed if we have a model of the process, namely HG_p(z). Depending on G_m(z), which is specified by the designer, different types of controllers will arise. Two of the most common are:

1. dead-beat controller and
2. Dahlin controller